Mixed Precision Symmetric Eigensolvers: Proof of Concept

The numerical solution of algebraic eigenvalue problems is a key technology underpinning many areas of computational science and engineering, including acoustics, aeronautics, control theory, fluid mechanics, population modelling, quantum physics, robotics, and structural engineering. In all these areas, the need for fast and numerically reliable solution of eigenvalue problems arises. The problems can be large so that time to solution can be unacceptably long.

Modern hardware (multicore processors and accelerators such as graphics processing units (GPUs)) increasingly supports half precision floating-point arithmetics. These low precisions provide new opportunities to considerably accelerate linear algebra computations.

The research in this "small grant" proposal is a proof of concept for the next generation of efficient and numerically stable eigensolvers that exploit the different arithmetic precisions of modern hardware while maintaining numerical stability. Our investigation concentrates on the symmetric eigenvalue problem, for which eigenvalues are real with a full set of orthonormal eigenvectors, but any advances will have direct impact on future algorithms for nonsymmetric eigenproblems, generalized eigenproblems, and the singular value decomposition.

The algorithms will be developed as prototypes in MATLAB, using simulated half precision. Their numerical stability will be analyzed as well as their efficiency in terms of arithmetic costs and communications costs so as to determine which one(s) should be fully implemented in state of the art numerical linear algebra libraries such as the freely available matrix algebra on GPU and multicore architectures (MAGMA) library.